On new sixth and seventh order iterative methods for solving non-linear equations using homotopy perturbation technique

Objectives This paper proposes three iterative methods of order three, six and seven respectively for solving non-linear equations using the modified homotopy perturbation technique coupled with system of equations. This paper also discusses the analysis of convergence of the proposed iterative methods. Results Several numerical examples are presented to illustrate and validation of the proposed methods. Implementation of the proposed methods in Maple is discussed with sample computations.


Introduction
The applications of non-linear equations of the type f (x) = 0 arise in various branches of pure and applied sciences, engineering and computing. In resent time, several scientists and engineers have been focused to solve non-linear equations numerically as well as analytically. In the literature, there are several iterative methods/algorithms available which are derived from different techniques such as homotopy, interpolation, Taylor's series, quadrature formulas, decomposition etc., and also available various modifications and improvements of the existing methods, and different hybrid iterative methods, see, for example [1, 4-7, 9-16, 28-32, 36-38]. In general, the roots of non-linear or transcendental equations cannot be expressed in closed form or cannot be computed analytically. The root-finding algorithms provide us to compute approximations to the roots, these approximations are expressed either as small isolating intervals or as floating point numbers. In this paper, we use the modified homotopy perturbation technique (HPT) to create a number of iterative methods for solving the given non-linear equations with converging order more than or equal to three. The given non-linear equations are expressed as an equivalent coupled system of equations with help of the Taylor's series and technique of He [4]. This enables us to express the given non-linear equation as a sum of linear and non-linear equations. The Maple implementation of the proposed algorithm is also discussed, and various Maple implementations for differential and transcendental equations are available in the literature, see, for example [17][18][19][20][21][22][23][24][25][26][27].
The rest of paper is organized as follows: Section recalls the preliminary concepts related to the topic; In Section , we present the methodology and steps involving in the proposed algorithms; Section discuses the analysis of convergence to show the order of proposed methods are more than or equal to three; Section presents several numerical examples to illustrate and validate the proposed methods/algorithms; and finally Section presents Page 2 of 15 Thota and Shanmugasundaram BMC Research Notes (2022) 15:267 the Maple implementation of the proposed algorithms with sample computations.

Preliminaries
In this paper, we consider the non-linear equation of the type Iterations techniques are a common approach widely used in various numerical algorithms/methods. It is a hope that an iteration in the general form of x n+1 = g(x n ) will eventually converge to the true solution α of the problem (1) at the limit when n → ∞ . The concern is whether this iteration will converge, and, if so, the rate of convergence. Specifically we use the following expression to represent how quickly the error e n = α − x n converges to zero. Let e n = α − x n and e n+1 = α − x n+1 for n ≥ 0 be the errors at n-th and (n + 1)-th iterations respectively. If two positive constants µ and p exist, and then the sequence is said to converge to α . Here p ≥ 1 is called the order of convergence, the constant µ is the rate of convergence or asymptotic error constant. This expression may be better understood when it is interpreted as |e n+1 | = µ|e n | p when n → ∞ . Obviously, the larger p and the smaller µ , the more quickly the sequence converges.
This paper focus on developing iterative methods/algorithms that are having the order of converges three, six and seven respectively. The following section presents the proposed methods using Taylor's series and modified HPT.

Main text
In this section, we present new iterative methods and its order of convergences with numerical examples, maple implementation and sample computations using maple mathematical software tool.

New iterative methods
We assume that α is an exact root of the equation (1) and let a be an initial approximation (sufficiently close) to α . We can rewrite the non-linear equation (1) using Taylor's series expansion as coupled system We have, from Newton's method, that From (4) and (5) Note that the equation (10) will play important role in the derivation of the iteration methods, see for example [2]. We use the technique of homotopy perturbation to develop the proposed iterative algorithms to solve the given non-linear equation (1). Using the HPT, we can construct a homotopy H(υ, p, m) : where p ∈ [0, 1] is embedding parameter and m ∈ R is unknown number. Clearly, from (11), we have Hence, the parameter p is monotonically increases on [0, 1]. The solution of equation (11) can be expressed as a power series in p Now the approximate solution of (1) is One can express the equation (11), as follows, by expanding T(x) using Taylor's series expansion around x 0 , By Putting (12) in (14), we get By comparing the coefficients of powers of p, we get . . are obtained as follows. From (16), we have From (17) and (20), we have From the assumption x 2 = 0 and from (19), we get From (6), (10) and (9), we have The approximate solution is obtained as This formulation allows us to form the following iterative methods.
Hence, for a given x 0 , we have the following iterative formula to find the approximate solution x n+1 . Hence, for a given x 0 , we have the following iterative schemes to find the approximate solution x n+1 .

Algorithm 3 For i = 3 , we have
Hence, for a given x 0 , we have the following iterative formula to find the approximate solution x n+1 .

Order of convergence
In this section, we show, in the following theorems, that the orders of converges of Algorithms 1, 2 and 3 are three, six and seven respectively. Let I ⊂ R be an open interval. To prove this, we follow the proofs of [9, Theorem 5, Theorem 6].
Theorem 2 Let f : I → R . Suppose α ∈ I is a simple root of (1) and θ is a sufficiently small neighborhood of α . Let f ′′ (x) exist and f ′ (x) � = 0 in θ . Then the iterative formula (28) given in Algorithm 1 produces a sequence of iterations {x n : n = 0, 1, 2, . . .} with order of convergence three. x

Let
Since α is a root of f(x), hence f (α) = 0 . One can compute that Hence the Algorithm 1 has third order convergence, by Theorem 1.
One can also verify that the order of convergence of Algorithm 1 as in the following example.

Example 1
Consider the following equation. It has a root α = √ 30 . We show, as discussed in proof of Theorem 2, that the Algorithm 1 has third order convergence.

Following Theorem 2, we have
Now Hence, by Theorem 2, the Algorithm 1 has third order convergence.
Theorem 3 Let f : I → R . Suppose α ∈ I is a simple root of (1) and θ is a sufficiently small neighborhood of α . Let f ′′ (x) exist and f ′ (x) � = 0 in θ . Then the iterative formula (29) given in Algorithm 2 produces a sequence of iterations {x n : n = 0, 1, 2, . . .} with order of convergence six.

Let
Since α is a root of f(x), hence f (α) = 0 . One can compute that Hence the Algorithm 2 has sixth order convergence, by Theorem 1.
We can also verify the order of convergence of Algorithm 2 as in the following example.

Example 2
Consider the equation (31). Using Theorem 3, similar to Example 1, we have Hence, by Theorem 3, the Algorithm 2 has sixth order convergence.

Now, we can check that
Theorem 4 Let f : I → R . Suppose α ∈ I is a simple root of (1) and θ is a sufficiently small neighborhood of α . Let f ′′ (x) exist and f ′ (x) � = 0 in θ . Then the iterative formula (30) given in Algorithm 3 produces a sequence of iterations {x n : n = 0, 1, 2, . . .} with order of convergence seven.

Proof
Let Since α is a root of f(x), hence f (α) = 0 . One can compute that Hence the Algorithm 3 has seventh order convergence, by Theorem 1.
Again, one can verify the order of convergence of Algorithm 3 using the following example.

Example 3
Consider the equation (31). Following Theorem 4, similar to Example 1 and Example 2, we have

Numerical example
This section presents several numerical examples to illustrate the proposed algorithms, and comparisons are made to confirm that the proposed algorithms give solution faster than existing methods.

Consider a non-linear equation
Suppose the initial approximation is x 0 = 2 with tolerance error 10 −10 correct to ten decimal places. Following the proposed algorithms (in equations 28, 29 and 30), we have Iteration-1 using Algorithm 1: Iteration-2 using Algorithm 1: Now,

Pseudo code
Input: Given f(x); initial approximation x[0]; tolerance ǫ ; correct to decimal places δ ; maximum number of iterations n. Output

Maple code
We present the maple code of the proposed algorithms as follows, and sample computations presented in Section. Similarly, one can apply the Algorithm 3 using Maple code.

Conclusion
In this paper, we presented three iterative methods of order three, six and seven respectively for solving nonlinear equations. With the help of modified homotopy perturbation technique, we obtained coupled system of equations which gives solution faster than existing methods. The analysis of convergence of the proposed iterative methods are discussed with example for each proposed method. Maple implementations of the proposed methods are discussed with sample sample computations. Numerical examples are presented to illustrate and validation of the proposed methods.